= Neumann cosine modes of a one-dimensional wave equation
{c}
{title2=$\psi(x,t)=A_0+B_0t+\sum_{n\geq1}[A_n\cos(cn\pi t/L)+B_n\sin(cn\pi t/L)]\cos(n\pi x/L)$}
For the <wave equation> $\psi_{tt}=c^2\psi_{xx}$ on $0<x<L$, impose homogeneous <Neumann boundary conditions> $\psi_x(0,t)=\psi_x(L,t)=0$, where $c,L>0$. <Separation of variables> gives $X''+\lambda X=0$ and $T''+c^2\lambda T=0$. <Integration by parts> gives $\int_0^L(X')^2dx=\lambda\int_0^LX^2dx$, so every nontrivial spatial <eigenfunction> has $\lambda\geq0$.
For $\lambda=0$, $X$ is constant and $T=A_0+B_0t$. For $\lambda>0$, the boundary conditions select $\lambda_n=(n\pi/L)^2$ and $X_n=\cos(n\pi x/L)$. The <superposition principle> gives the displayed expansion when the initial data have sufficient regularity. The constant <zero mode> must be retained: homogeneous <Neumann boundary conditions> do not force the spatial average of the displacement to vanish.
With zero initial <velocity>, all $B_n$ vanish, including $B_0$. Initial displacement $a+b\cos(n\pi x/L)$ then evolves as $a+b\cos(cn\pi t/L)\cos(n\pi x/L)$, a <standing wave> above a constant offset.
Back to article page
